Integrating signals, and some moonbounce history

(This is an adapted version of part of an article that I wrote for the Dutch
amateur radio magazine Electron, July 2015.)

Integrating signals

When we want to fish a weak but long-lived (or repetitive) signal from the noise,
we do that by "integration".
That means something like adding or averaging the signals received at different times.
There are two fundamentally different ways of integration: coherent
and non-coherent.
Coherent integration means we first average the signal and then
determine their strength (power).
Non-coherent integration is the opposite: first determine the strength (power),
and then averages those strengths.
The difference is illustrated in the figure.
At top left, we see the received signal: in this case 100 periods (of which 5 shown)
of some sine signal, with so much noise added that the sine is hard to recognize.
At the right of this, coherent integration is illustrated, and at the bottom
non-coherent integration.

Coherent integration means we shift those 100 sine periods on top of each other,
add them, and divide by 100 (i.e., take the average).
This leads to the figure at the right, in which the sine is easily recognized.
This is because the sine signal is the same in each period, and thus becomes 100
times as big when we add 100 instances.
The noise signals is different during each period of the sine, and when added,
will therefore sometimes get larger and sometimes get weaker;
in case of adding 100 instances, the noise will become, on average, only 10 times
as large (that 10 is the square root of 100), and after division by 100 it therefore
becomes 10 times smaller.
The net result is that integrating coherently by a factor of 100, as done here,
the signal/noise ratio, measured as a voltage, improves 10-fold;
measured as a power this is a factor of 100.

In fact, this operation works as a filter.
Consider what would happen if we had a sinewave of a slightly lower or slightly higher
frequency: then in the consecutive intervals, this sine would not have the same
phase, and would not add constructively.
This filter operation explains that we get rid of noise: noise is a mixture of
frequencies, many of which will get attenuated by this filter.
As we add more time slots, the filter effect leads to a larger phase difference
between the first and the last timeslot (for frequencies other than the one
whose period matching the slot length), and thus the filter gets narrower, leaving
less noise.

The bottom half of the figure illustrates non-coherent integration.
For each of the 100 timeslots, we first measure the power.
This power is partially contributed by the desired signal (same in every timeslot),
and partially by the noise (varies between timeslots).
Because of the noise contribution, the measure power is not the same every time:
we see rather strongly fluctuating numbers.
To determine the power more precisely, we can compute the average of these 100
numbers: that reduces the fluctuation by a factor of 10 (again the square root of 100).

As we integrate non-coherently for a longer time, we thus get an increasingly precise
estimate of the total power, i.e., the power of the desired signal plus
the noise power.
To determine whether the desired signal is really there, we have to perform the same
operation on another received signal, in which we are sure the desired signal is
not present but which has the same noise level,
e.g. received on a neighbouring frequency or at a different time.
Then we can determine the power of the latter signal too, and check whether there
is a significant difference.
The longer we integrate, the more precise the power estimate is (fluctuations have
been averaged out more), and thus the smaller the power differences (i.e., the
desired signal) we can reliably detect.

I can now reveal that for the above figure, I created a signal in which the
noise power was 100 units (let's say mW), and the sine signal 200 mW.
We see that the non-coherent integration indeed yields a decent estimate of the
total power, which is much closer to the theoretical value (300 mW)
than the measurements over the individual periods.
And in the coherent integration we see indeed that most of the noise power is gone.
Theoretically, the coherent integration should give 201 mW, namely 200 mW for the
sine signal, and, because of the filter, 1/100 of 100 mW of noise.
The fact that we get 211.64, is again a consequence of the random nature of the noise,
and thus of the numbers;
if one repeats the experiment with different but equally strong noise, one might
find e.g. 195.

If we compare both methods, then coherent integration seems best, since it
actually removes noise.
However, in practice both methods are used, often in combination.
The reason is that, as pointed out above, coherent integration acts as a filter,
and this filter becomes narrower if we integrate longer.
If the frequency of the desired signal is not very stable, one cannot integrate
coherently for very long: between beginning and end of the integration the phase
of the signal might change too much.
The optimal thing then is to integrate coherently over a time which matches the
stability of the desired signal, and if that is not enough, use non-coherent
integration to integrate further.

This is done e.g. in decoding WSPR signals.
The individual tones, each lasting 0.7 s, are integrated coherently,
but combining the information from all the tones is done non-coherently.
This is because propagation on HF is not very stable; a signal can easily be
smeared out over a Hz of bandwidth.
On LF, propagation is more stable, and there WSPR-15 is often used:
this is a variant which is 7.5 times slower than normal WSPR.
And on VLF, one can integrate coherently for hours, as also mentioned in
the previous installment.

Another place where coherent and non-coherent integration show up, is in the
waterfall displays in SDR software.
Often, such software has a control for the "FFT size": this is in fact a measure
for how long the coherent integration is.
After that, the FFT output can be averaged further: that is non-coherent integration.
Some software (such as Rocky)
can even make the averaging time depend on what is being received: for strong signals
the averaging time is short (so fast morse signals can be distinguished), while
for weak signals it is long (so even very weak signals do become visible).

A typical application where only non-coherent integration can be used,
is radio astronomy.
The signals to be received there are (in most cases) wide-band noise, so their
presence can only be detected by noticing that from some direction in the sky
the received noise (consisting of the receiver's own noise and the
noise received from the sky, if any) is a bit stronger than from some other direction.
The longer one integrates, the more precise the noise level can be measured, and
thus the smaller the differences are that still can be recognized.

A rather unusual integrator for moonbounce

How does one build an integrator (coherent or not) ?
In analog electronics, using a capacitor seems obvious, which one charges with
a current proportional to the received signal.
Unfortunately, capacitors are not perfect; they leak, and also the electronics
connected to it will draw some current from them.
Nowadays therefore, integration is almost always done digitally, usually in software.
Digital adders work with exact numbers, expressed in ones and zeros, and thus
integrate without leaking.

In the past, digital adders were not yet available.
In 1946 Zoltán Bay from Hungary was one of the first who managed to detect radio
signals reflected by the moon.
His receiver was not really sensitive enough, and therefore he could not directly
measure his own echoes off the moon.
He needed to "integrate" many consecutive transmissions to lift the
echoes out of the noise.
For this, he came up with the rather original solution illustrated here (pictures
copied from his 1947 article).
His setup consists of a glass structure containing salty water, an anode electrode
at the top, and a series of cathode electrodes at the bottom.
When a current is fed through the water, from the anode to one of the cathodes,
the water is electrolysed: oxygen gas is produced at the anode, and hydrogen gas
at the cathode.
Being a gas, the hydrogen will bubble up into the glass tube above the cathode.
The quantity of gas accumulating at the cathode, is proportional to the total
amount of electrical charge that has flown to that cathode:
it's a perfect integrator, assuming the glass is gas-tight.
Such a thing is also called a "coulometer", after the unit of electrical
charge, the Coulomb.
Furthermore, Zoltán Bay used a rotating switch to
on the one hand briefly activate the transmitter every 3 seconds, and on the
other hand connect the receiver periodically, briefly, to each of the cathodes.
Since the time needed for the signal to travel to the moon and back is the same
every time, the echo wil always return at the moment the same cathode is connected.
It is therefore to be expected that at one cathode more gas will be generated than
at the others; and this is indeed what he saw, thus proving that his signal had
indeed been reflected by the moon!

Zoltán Bay's report
(Z. Bay: Reflection of microwaves from the moon, Hungarica Acta Physica, Jan. 1947)
of these experiments is still quite readable nowadays.
He worked on 120 MHz, using an antenna consisting of 36 dipoles in front of a reflector,
with some 3 to 4 kW of transmit power.
He started his experiments already during World War 2, and had to partially rebuild
his setup after the war.
Perhaps that is why he was not the first person to achieve moonbounce: an American
team beat him by just one month in 1946.
That American team had much better frequency stability, and could therefore
use a much more narrow-band receiver, which effectively is a coherent
integrator.
Therefore, they could detect the echoes of the individual transmissions,
without needing non-coherent integration of successive transmissions.

The very first moonbounce

The honour of the real first moonbounce however goes neither to the Hungarian,
nor to the American team, but to a German team, albeit unintentionally
(see this webpage
or the more detailed article by DK2ZF in CQ-DL 7/1979, p. 328, reproduced
here).
Telefunken was testing a radar in the autumn of 1943 on the island of Rügen,
with 120 kW of transmit power on 564 MHz, and a 45 m2 antenna.
On the receiver they saw an unexpected signal, which strangely only appeared
some 2 seconds after switching on the transmitter, while all other normal signals
did appear immediately.
Furthermore, this unexpected signal was only present of the antenna was rotated
into a specific direction, and appeared only 2 seconds after the antenna had been
pointed into that direction.
You guessed it: that was precisely the direction in which the moon stood...

Text and pictures on this page (except for the two pictures from Z. Bay's paper)
are copyright 2015, P.T. de Boer, pa3fwm@amsat.org .
Republication is only allowed with my explicit permission.